102 V NORMED SPACES

4. SUBS PACES AND FINITE PRODUCTS OF NORMED SPACES

Let E be a nonned space, F a vector subspace of E (i.e. a subset such
that jc e F and y e F imply ax + py e F for any pair of scalars a, /?); the
restriction to F of the norm of E is clearly a norm on F, which defines on
F the distance and topology induced by those of E. When talking of a
" subspace" of E, we will in general mean a vector subspace with the induced
norm. If E is a Banach space, any closed subspace F of E is a Banach space
by (3.14.5); conversely, if a subspace F of a normed space E is a Banach
space, F is closed in E by (3.14.4).

(5.4.1) If F is a vector subspace of a normed space E, its closure F in E is a
vector subspace.

By assumption, the mapping (x9y)-+x+y of E x E into E maps F x F
into F, hence maps F x F into F, by (3.11.4); as Fx F = F x F by (3.20.3),
the relations jc e F, y e F imply x + y e F. Using the continuity of (A, x) -+lx,
we similarly show that x e F implies Ax e F for any scalar L

We say that a subset A of a normed space E is total if the (finite) linear
combinations of vectors of A form a dense subspace of E; we say that a
family (x^ is total if the set of its elements is total.

Let El5 E2 be two normed spaces, and consider the product vector
space E = E! x E2 (with (xi9 x2) + 0>i, y2) = (*i + J>i, x2 + y2) and
k(xi9x2) = (Ax1? Ax2)). It is immediately verified that the mapping
(*!, x2) -* sup (H^ill, IIX2II) is a norm on E, which defines on E the distance
corresponding to the distances on El5E2, and therefore the topology of
the product space.Ej_ x E2 as defined in Section 3.20. The "natural" in-
jections xl -> (%!, 0), x2 -+ (0, x2) are linear isometries of Et and E2 respect-
ively onto the closed subspaces EJ = Ex x {0}, E2 = {0} x E2 of E (3.20.11),
and E is the direct sum of its subspaces E'x, E2, which are often identified
to E!, E2 respectively.

Conversely, suppose a normed space E is a direct sum of two vector
subspaces Fl5 F2; each xeE can be written in a unique way x =PI(X) +p2(x)9
with PI(X) e Fl3 p2(x) e F2, and pi9p2 are linear mappings of E into F1? F2
respectively (the "projections" of E onto Fi9 F2). The "natural" mapping
(yi> Ja) ~* J7! + 72 is a linear bisection of the product space F! x F2 onto E,
which is continuous (by (5.1.5)), but not necessarily bicontinuous (see Section
6.5, Problem 2). series of numbers ^= 0. Show that